Problem 48 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{7}{16 a^{2} b}+\frac{3 a}{20 b^{2}} $$

Step-by-Step Solution

Answer

\(\frac{35b + 12a^3}{80a^2b^2}\)

Step 1: Identify the Common Denominator

The denominators in the rational expressions are \(16a^2b\) and \(20b^2\). To find a common denominator, we need the least common multiple (LCM) of these terms. The LCM of 16 and 20 is 80, and the LCM of the variables is \(a^2b^2\). Thus, the common denominator is \(80a^2b^2\).

Step 2: Rewrite Each Fraction with the Common Denominator

Convert each fraction to have the common denominator \(80a^2b^2\). For \(\frac{7}{16a^2b}\), multiply both the numerator and denominator by \(5b\):\[\frac{7}{16a^2b} \times \frac{5b}{5b} = \frac{35b}{80a^2b^2}\]For \(\frac{3a}{20b^2}\), multiply both the numerator and denominator by \(4a^2\):\[\frac{3a}{20b^2} \times \frac{4a^2}{4a^2} = \frac{12a^3}{80a^2b^2}\]

Step 3: Add the Fractions

Now that both fractions have the common denominator \(80a^2b^2\), add the numerators together:\[\frac{35b}{80a^2b^2} + \frac{12a^3}{80a^2b^2} = \frac{35b + 12a^3}{80a^2b^2}\]

Step 4: Simplify the Resulting Fraction

Check the numerator for any common factors. In this case, 35 and 12 have no common factors, and \(b\) and \(a^3\) share no variables. Therefore, \(35b + 12a^3\) cannot be simplified further with respect to the denominator \(80a^2b^2\). So, the expression \(\frac{35b + 12a^3}{80a^2b^2}\) is already in simplest form.

Key Concepts

Common Denominator

In order to add or subtract rational expressions, a common denominator is needed. This is similar to adding or subtracting ordinary fractions. The common denominator is essentially a shared multiple of all denominators present in the problem. To find it, we determine what "common theme" exists among the denominators to make them alike.

Consider the rational expressions \(\frac{7}{16a^2b}\) and \(\frac{3a}{20b^2}\). Here, the denominators \(16a^2b\) and \(20b^2\) are different. We must find a common denominator that both denominators divide evenly into, ensuring both fractions are on the same page, so to speak.

• This "common ground" comes from the Least Common Multiple (LCM) of both denominators.
• The LCM accounts for both numerical coefficients (in this case, 16 and 20) and all variables (\(a\) and \(b\)) raised to the necessary power.
• Once found, each term must be adjusted to fit this common denominator.

Getting comfortable with finding and using a common denominator takes practice but makes complex algebraic operations much simpler.

Least Common Multiple

The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. While often taught in the context of simple numbers, it plays a crucial role in algebra when dealing with polynomials and rational expressions.

When finding the common denominator in rational expressions, like in our example with \(16a^2b\) and \(20b^2\), the LCM allows these distinct terms to be expressed with a uniform denominator. Here's how it's done:

• Find the LCM of the constants: For 16 and 20, the LCM is 80. This is found by identifying the smallest number divisible by both.
• Address the variables separately: For \(a^2\) and \(b\) compared with \(b^2\), the LCM would be \(a^2b^2\). This includes the highest powers of all variables present.

So, the common denominator is \(80a^2b^2\). Finding the LCM ensures all parts of the expression have the same basis, paving the way for addition or subtraction.

Simplifying Fractions

Once fractions are added or subtracted, the resulting expression should be simplified, if possible. This means reducing the expression to its lowest terms by eliminating unnecessary parts, usually through finding common factors.

In our simplified result \(\frac{35b + 12a^3}{80a^2b^2}\), simplifying involves several steps:

• Evaluate the numerator: Does it have factors that match any part of the denominator? In this case, \(35b\) and \(12a^3\) do not share factors with the bottom \(80a^2b^2\).
• Check for common factors: Since \(35\) and \(12\) have no common numeric factors, and \(b\) and \(a^3\) share no variables, we confirm that further simplification isn't possible.

Thus, knowing when fractions are already in their simplest form is as important as knowing how to simplify them. Knowing the structure of both the numerator and the denominator is key to correctly simplifying expressions.